scholarly journals On the cohomology of the space of seven points in general linear position

2020 ◽  
Vol 6 (4) ◽  
Author(s):  
Olof Bergvall
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Finite Fields ◽  
General Linear ◽  
Cohomology Groups ◽  
Point Counts

AbstractWe determine the cohomology groups of the space of seven points in general linear position in the projective plane as representations of the symmetric group on seven elements by making equivariant point counts over finite fields. We also comment on the case of eight points.

scholarly journals A Bound on Permutation Codes

10.37236/2929 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
Jürgen Bierbrauer ◽  
Klaus Metsch
Keyword(s):  
Projective Plane ◽  
Symmetric Group ◽  
Minimum Distance ◽  
Main Part ◽  
Latin Squares ◽  
Permutation Codes ◽  
Permutation Code ◽  
Mutually Orthogonal Latin Squares ◽  
Hamming Metric ◽  
Completion Theorem

Consider the symmetric group $S_n$ with the Hamming metric. A  permutation code on $n$ symbols is a subset $C\subseteq S_n.$ If $C$ has minimum distance $\geq n-1,$ then $\vert C\vert\leq n^2-n.$ Equality can be reached if and only if a projective plane of order $n$ exists. Call $C$ embeddable if it is contained in a permutation code of minimum distance $n-1$ and cardinality $n^2-n.$ Let $\delta =\delta (C)=n^2-n-\vert C\vert$ be the deficiency of the permutation code $C\subseteq S_n$ of minimum distance $\geq n-1.$We prove that $C$ is embeddable if either $\delta\leq 2$ or if $(\delta^2-1)(\delta +1)^2<27(n+2)/16.$ The main part of the proof is an adaptation of the method used to obtain the famous Bruck completion theorem for mutually orthogonal latin squares.

scholarly journals Analysis of Some Properties from Construction of Finite Projective Planes of Order 2 and 3

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 131
Author(s):  
Vira Hari Krisnawati ◽  
Corina Karim
Keyword(s):  
Automorphism Group ◽  
Projective Plane ◽  
Symmetric Group ◽  
Block Design ◽  
Finite Projective Plane ◽  
Projective Planes ◽  
Steiner System ◽  
Incidence Relation ◽  
Finite Projective Planes ◽  
Set Of Points

<p class="abstract"><span lang="IN">In combinatorial mathematics, a Steiner system is a type of block design. Specifically, a Steiner system <em>S</em>(<em>t</em>, <em>k</em>, <em>v</em>) is a set of <em>v</em> points and <em>k</em> blocks which satisfy that every <em>t</em>-subset of <em>v</em>-set of points appear in the unique block. It is well-known that a finite projective plane is one examples of Steiner system with <em>t</em> = 2, which consists of a set of points and lines together with an incidence relation between them and order 2 is the smallest order.</span></p><p class="abstract"><span lang="IN">In this paper, we observe some properties from construction of finite projective planes of order 2 and 3. Also, we analyse the intersection between two projective planes by using some characteristics of the construction and orbit of projective planes over some representative cosets from automorphism group in the appropriate symmetric group.</span></p>

scholarly journals Counting the number of trigonal curves of genus 5 over finite fields

2020 ◽  
Vol 208 (1) ◽  
pp. 31-48
Author(s):  
Thomas Wennink
Keyword(s):  
Finite Field ◽  
Projective Plane ◽  
Euler Characteristic ◽  
Finite Fields ◽  
Moduli Space ◽  
Plane Curves ◽  
Sieve Method ◽  
Main Application ◽  
Trigonal Curves

AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

scholarly journals General Linear and Functor Cohomology Over Finite Fields

1999 ◽  
Vol 150 (2) ◽  
pp. 663 ◽  
Author(s):  
Vincent Franjou ◽  
Eric M. Friedlander ◽  
Alexander Scorichenko ◽  
Andrei Suslin
Keyword(s):  
Finite Fields ◽  
General Linear

scholarly journals Abelian p-subgroups of the general linear group

1970 ◽  
Vol 11 (3) ◽  
pp. 257-259 ◽  
Author(s):  
J. T. Goozeff
Keyword(s):  
Finite Field ◽  
Linear Group ◽  
Finite Fields ◽  
Abelian Subgroup ◽  
General Linear Group ◽  
General Linear ◽  
Normal Abelian Subgroup ◽  
Abelian Subgroups

A. J. Weir [1] has found the maximal normal abelian subgroups of the Sylow p-subgroups of the general linear group over a finite field of characteristic p, and a theorem of J. L. Alperin [2] shows that the Sylow p-subgroups of the general linear group over finite fields of characteristic different from p have a unique largest normal abelian subgroup and that no other abelian subgroup has order as great.

scholarly journals A theorem on power series with applications to classical groups over finite fields

2001 ◽  
Vol 64 (1) ◽  
pp. 121-129
Author(s):  
Andrew J. Spencer
Keyword(s):  
Power Series ◽  
Finite Fields ◽  
Generating Functions ◽  
General Linear ◽  
Classical Groups ◽  
Orthogonal Groups

For some of the classical groups over finite fields it is possible to express the proportion of eigenvalue-free matrices in terms of generating functions. We prove a theorem on the monotonicity of the coefficients of powers of power series and apply this to the generating functions of the general linear, symplectic and orthogonal groups. This proves a conjecture on the monotonicity of the proportions of eigenvalue-free elements in these groups.

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